Finite-to-finite universal quasivarieties are Q-universal

Let K be a quasivariety of algebraic systems of finite type. K is said to be universal if the category G of all directed graphs is isomorphic to a full subcategory of K. If an embedding of G may be effected by a functor F : G --> K which assigns a finite algebraic system to each finite graph, then K is said to be finite-to-finite universal. K is said to be e-universal if, for any quasivariety M of finite type, L(M) is a homomorphic image of a sublattice of L(K), where L(M) and L(K) are the lattices of quasivarieties contained in M and K, respectively. We establish a connection between these two, apparently unrelated, notions by showing that if K is finite-to-finite universal, then K is e-universal. Using this connection a number of quasivarieties are shown to be e-universal.